Problem 15 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

The rate constant, the activation energy and the Arrhenius parameter of a chemical reaction at \(25^{\circ} \mathrm{C}\) are \(3.0 \times 10^{-4} \mathrm{~s}^{-1}, 104.4 \mathrm{~kJ} \mathrm{~mol}^{-1}\) and \(6.0\) \(\times 10^{14} \mathrm{~s}^{-1}\) respectively. The value of the rate constant as \(T \rightarrow \infty\) is, (a) \(2.0 \times 10^{18} \mathrm{~s}^{-1}\) (b) \(6.0 \times 10^{14} \mathrm{~s}^{-1}\) (c) infinity (d) \(3.6 \times 10^{30} \mathrm{~s}^{-1}\)

Step-by-Step Solution

Answer

As \( T \to \infty \), the rate constant approaches \( 6.0 \times 10^{14} \mathrm{~s}^{-1} \).

Step 1: Understanding the Arrhenius Equation

The Arrhenius equation is given by \( k = A e^{-\frac{E_a}{RT}} \) where \( k \) is the rate constant, \( A \) is the Arrhenius parameter (or pre-exponential factor), \( E_a \) is the activation energy, \( R \) is the gas constant (8.314 J/mol·K), and \( T \) is the temperature in Kelvin.

Step 2: Determine the Limit as Temperature Approaches Infinity

As \( T \rightarrow \infty \), the exponential term \( e^{-\frac{E_a}{RT}} \) approaches 1 because the exponent approaches 0 (since \( E_a \) is divided by an infinitely large number). Thus, \( k \) approaches \( A \).

Step 3: Identifying the Correct Answer

Given that the Arrhenius parameter \( A \) is \( 6.0 \times 10^{14} \text{ s}^{-1} \), as \( T \rightarrow \infty \), the rate constant \( k \) approaches \( 6.0 \times 10^{14} \text{ s}^{-1} \). Thus, the correct answer is option (b).

Key Concepts

Rate Constant

In the world of chemistry, the rate constant, often denoted as \( k \), plays a central role in determining how fast a reaction proceeds. It is a numerical value that reflects the speed of a chemical reaction at a specific temperature. Unlike concentrations of reactants or products, the rate constant is a fixed value for a given reaction at a particular temperature.

• It is influenced by the nature of the reactants involved and the conditions under which the reaction takes place.
• The rate constant is crucial because it helps predict how quickly products will form or reactants will disappear over time.
• Mathematically, it appears in rate laws, which describe how the rate of a reaction depends on the concentration of its reactants.

Understanding the rate constant gives chemists the ability to control and optimize industrial and laboratory processes by adjusting temperatures or using catalysts to modify \( k \). The rate constant is not the only factor affecting reaction rates, but it encapsulates several underlying factors into one meaningful number.

Activation Energy

Activation energy, represented as \( E_a \), is a core concept needed to understand why some reactions are slow and others are fast. It is the minimum amount of energy required to initiate a chemical reaction. Think of it as the barrier that reactants must overcome to be transformed into products.

• Reactions with high activation energy tend to proceed slower than those with low activation energy, all else being equal.
• It reflects the strength of bonds that must be broken and formed during a chemical reaction.
• A common method to reduce activation energy is the use of catalysts, which provide an alternative reaction pathway with a lower activation energy.

Activation energy is key to controlling reaction rates and is closely linked to temperature and the rate constant through the Arrhenius equation, making it a fundamental parameter in chemical kinetics.

Temperature Dependence

Temperature has a significant impact on the rate of chemical reactions. As temperature increases, the kinetic energy of molecules rises, leading to more frequent and energetic collisions. This increases the likelihood of overcoming the activation energy barrier, leading to faster reactions.

• The Arrhenius equation captures this relationship, showing how the rate constant \( k \) changes with temperature.
• In practice, even a small increase in temperature can lead to a substantial increase in the rate constant, and vice versa.
• This temperature dependence is why many chemical processes are temperature-sensitive and need precise control.

In the limit of very high temperatures, as addressed in the original exercise, the exponential factor in the Arrhenius equation approaches 1, indicating that the rate constant approaches the Arrhenius pre-exponential factor \( A \). This demonstrates how potent the effect of temperature can be on reaction kinetics.

Chemical Kinetics

Chemical kinetics is the branch of chemistry that deals with understanding the rates of chemical reactions. It seeks to explain how different factors like concentration, temperature, and the nature of the reactants influence these rates. Chemical kinetics provides a deeper insight into reaction mechanisms by studying how reactions proceed on a molecular level.

• Kinetics involves both experimental measurements and theoretical calculations to understand reaction rates.
• It also considers the effects of catalysts, which can dramatically decrease activation energy and accelerate reaction rates.
• The field uses mathematical models, including the Arrhenius equation, to predict and explain the behavior of reacting substances.

Studying chemical kinetics equips scientists with the tools to innovate in fields like pharmaceuticals, environmental science, and materials engineering by optimizing and controlling chemical processes for improved yields and efficiencies.